lecture07 - MA1100 Lecture 7 Mathematical Proofs Parity of...

Info icon This preview shows pages 1–10. Sign up to view the full content.

View Full Document Right Arrow Icon
1 MA1100 Lecture 7 Mathematical Proofs Parity of integers Divisibility of integers Direct Proofs Proof by Contrapositive Chartrand: section 3.2, 3.3, 4.1
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Lecture 7 2 True Statements • Mathematical statements that are true are known as mathematical facts . • A proposition is a true mathematical statement that has a proof . • A proposition that is “ important ” is called a theorem . • A proposition that is used in the proof of a theorem is called a lemma . • A proposition that follows (easily) from a theorem is called a corollary . • There are true statements that do not have proofs e.g. definitions , axioms . Chartrand calls it “Result”
Image of page 2
Lecture 7 3 An Example If x is an even integer, then x 2 is divisible by 4. Proposition We want to give a proof for this (universal) conditional statement. We need to : know the definitions of even integers and divisibility use some basic facts about integers use algebraic manipulation
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Lecture 7 4 Definition Examples (Mathematical) 1. Even and odd integers 2.n is divisible by m A (mathematical) definition gives the precise meaning of a word or phrase that represents some object, property or other concepts.
Image of page 4
Lecture 7 5 Even integers 2 6 -14 Example = 2 ä 1 = 2 ä 3 = 2 ä (-7) An integer a is an even integer if there exists an integer n such that a = 2n . Definition In general, an even integer can be written in the form 2n where n is an integer
Image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Lecture 7 6 Odd integers 3 7 -13 Example = 2 ä 1 + 1 = 2 ä 3 + 1 = 2 ä (-7) + 1 An integer a is an odd integer if there exists an integer n such that a = 2n + 1 . Definition In general, an odd integer can be written in the form 2n + 1 where n is an integer
Image of page 6
Lecture 7 7 Even and odd integers Why do we need definitions for even and odd integers? Is 0 an even integer? Not to identify specific even and odd integers To describe general even and odd integers To be used in the proof of statements involving even and odd integers
Image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Lecture 7 8 Undefined Terms Some basic concepts remain undefined . Numbers, integers, … Addition, multiplication, … Point, line, … Examples
Image of page 8
Lecture 7 9 Divisibility Definition Let m and n be integers. We say m divides n if there exists an integer q such that n = mq .
Image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 10
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern